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The following problem classes are all convex optimization problems, or can be reduced to convex optimization problems via simple transformations: [7]: chpt.4 [10] A hierarchy of convex optimization problems. (LP: linear programming, QP: quadratic programming, SOCP second-order cone program, SDP: semidefinite programming, CP: conic optimization.)
A minimum spanning tree of a weighted planar graph.Finding a minimum spanning tree is a common problem involving combinatorial optimization. Combinatorial optimization is a subfield of mathematical optimization that consists of finding an optimal object from a finite set of objects, [1] where the set of feasible solutions is discrete or can be reduced to a discrete set.
A complexity class is a set of problems of related complexity. Simpler complexity classes are defined by the following factors: The type of computational problem: The most commonly used problems are decision problems. However, complexity classes can be defined based on function problems, counting problems, optimization problems, promise ...
The hierarchy has classes as nodes and inheritance relationships as links. Classes in the same level are more likely to be associated than classes in different levels. The levels of this hierarchy are called layers or levels of abstraction. Example (Simplified Objective-C 2.0 code, from iPhone SDK):
MATLAB (an abbreviation of "MATrix LABoratory" [18]) is a proprietary multi-paradigm programming language and numeric computing environment developed by MathWorks.MATLAB allows matrix manipulations, plotting of functions and data, implementation of algorithms, creation of user interfaces, and interfacing with programs written in other languages.
Note how the use of A[i][j] with multi-step indexing as in C, as opposed to a neutral notation like A(i,j) as in Fortran, almost inevitably implies row-major order for syntactic reasons, so to speak, because it can be rewritten as (A[i])[j], and the A[i] row part can even be assigned to an intermediate variable that is then indexed in a separate expression.
Type classes permit multiple type parameters, and so type classes can be seen as relations on types. [6] For example, in the GHC standard library, the class IArray expresses a general immutable array interface. In this class, the type class constraint IArray a e means that a is an array type that contains elements of type e.
If all the hard constraints are linear and some are inequalities, but the objective function is quadratic, the problem is a quadratic programming problem. It is one type of nonlinear programming. It can still be solved in polynomial time by the ellipsoid method if the objective function is convex; otherwise the problem may be NP hard.